The Golden Sheets
those who may have missed class when these were handed out, here are the
Golden Sheets. To print these pages, work box by box and
block/copy/paste each set of contents in each box into a word processing
document. If you have any trouble with the golden box following along
with the print, it can help to copy only to the next to last character
(i.e.: leave out the period at the end of the last sentence).
We both urge you to
attend both the class and the tutorial and to do these exercises
religiously. Symbolic logic does not get learned by osmosis, there are
no Coles Notes for it, and there isn't a video you can rent at
Blockbuster to sum it all up. You will only learn it by doing it, by
asking questions about it, by working through it and discussing it with
The Golden Sheets,
GOLDEN SHEETS, SET 1
Categorical Propositions can be classified
according to 2 criteria
(a) Affirmative S
(b) Negative S
is not P
All S are P/ No S is P
Some S are P/ Some S are not P
2) Combining Quality and Quantity
There are 4 types:
All S are P
All human beings are mortal
No S are P
No human being is immortal
Some S are P
Some human beings are mortal
Some S are not P
Some human beings are not immortal
Golden Sheets, Set 1, page 2
DISTRIBUTION OF TERMS
(Moris Engel The Chain of Logic, Ch2)
Before considering other types of immediate inferences,
we need to look at a technical matter concerning the four
categorical propositions. Called distribution of terms, it is
concerned with three points: the classes designated by the terms,
whether or not those classes are occupied, and to what extent they
A reference is made in the four categorical propositions
regarding the classes designated by their terms. We want to know
whether the reference is to the whole class or only to part of the
class. If it is to the whole class, then the class is said to be
distributed; if the reference is only to part of the class, the
class is undistributed.
The A proposition asserts that every member of the subject
class is a member of the predicate class. Since reference is made to
every member of the subject class, the subject term is
distributed. But is reference being made to every member of the
predicate class? The answer is no. If I say, for example, “I am not
asserting that only artists are eccentric, nor am I saying
that artists make up the whole class of eccentric people. I
am only asserting that if a person is an artist, he is eccentric.
But other people may be eccentric too, so the predicate term of the
A proposition is undistributed.
We can now see more clearly why the A proposition does not
convert simply, but rather only by limitation, to an I. The A
proposition has a distributed subject and an undistributed
predicate. In conversion, the predicate becomes the subject, and
what was undistributed in the original is now distributed. Such an
inference is invalid, since this would be equivalent to jumping from
a knowledge of some things to a presumed knowledge of all
Like the A, the E proposition’s quantifier makes reference
(albeit in a negative way) to every member of the subject class.
Unlike the A, however, the E states that not a single member of the
S class is a member of the P class; thus the reference is to the
whole of the predicate class. How could it maintain this to be so
unless the whole of that class was surveyed and no S was
found in it? Therefore, the predicate term in the E proposition is
distributed. And because both terms are distributed, that
proposition, unlike the A, converts simply.
In the I proposition, the quantifier makes it clear that only
some members of the subject class are in discussion, so the subject
is undistributed. But is the predicate term similarly undistributed?
The answer is yes, since reference is being made here only to some
members of that class, not to the whole of it. In a proposition like
“Some men are wealthy,” we need to identify only those members of
the predicate class who are also members of the subject class; we
are not concerned about the rest of the P class, which may be
coextensive with other types of subject classes (for example, women
who are wealthy).
Because both classes in the I proposition share the same kind
of distribution, it can be converted simply. In interchanging the
subject and predicate terms, as is required by the process of
conversion, we are not going from an undistributed term (involving
knowledge only about “some”) to a distributed one (involving
knowledge about “all”), as we would if we attempted to convert the A
As in the I, the quantifier “some” in the O proposition
indicates that reference is being made to only a part of that class.
The subject term of the O proposition is therefore undistributed. Is
the predicate term also undistributed? The answer is no. The P term
is distributed because if something is excluded from a class,
the whole of the class is necessarily involved. How would we know
that a certain S is not a member of a certain P class unless we had
surveyed that whole P class and failed to find it there?
Because of its distribution, the O proposition cannot be
validly converted. To convert it would involve transposing the S
term to the P position, a position that is distributed, and the data
of the original does not justify this. It would mean using the knowledge of only some to
claim knowledge about all.
The following table
illustrates the distribution situation for subject and predicate
terms in each of the four categorical propositions:
symmetry between the A and O forms and between the E and I forms in
this representation. The subject of distribution can be difficult.
The distribution of the subject term, however, is indicated clearly
by the quantifier and should therefore offer no problems. As far as
the predicate term is concerned, it may be helpful to remember that
it is distributed only in the negative propositions (the E and O).
Golden Sheets, Set 1, page 3
DETERMINATION OF VALID MOODS OF THE SYLLOGISM
Possible Moods based on A, E, I, and O combinations
ii) Each of these 64 moods can appear in
each of the 4 figures. Therefore, the total number of possible moods
is 64 x 4 = 256 possible cases. However, not all of them are valid
moods. In order to determine the number of valid moods, we need to
look into the 64 possible moods above, and then apply the 7 rules of
(from Engel, Morris (1987)
The Chain of Logic, Chapter Two, Section 5)
Rule 1: The middle term must be
distributed at least once.
Rule 2: If a term is distributed in the conclusion, it must be
distributed in the premise.
Rule 3: From two negative premises no conclusion follows.
Rule 4: If one premise is negative, the conclusion must be negative;
if the conclusion is
negative, one premise must be negative
Rule 5: If a syllogism is to be valid, it can have only three terms.
Rule 6: From 2 particular premises, no conclusion follows.
Rule 7: If 1 premise is particular, the conclusion must be
One rule can be applied immediately: rule
number 3. If so, then we can see immediately that the cases of EE,
EO, OE, and OO are invalid. We may therefore cross out as invalid
the entire fourth column in the table above.
iii) Complete the table; then apply as many rules as you can
determine the valid moods. If
a mood does not violate the rules, it is valid, but if one rule
is violated, the mood is invalid.
Golden Sheets, Set 1, page 4
Rewrite the following syllogistic arguments in
standard form (putting the major premise first, the minor premise
second and the conclusion last). Each has some missing component and
you should supply it and indicate it with an arrow <--------
1. Example: Since logic is clear, it is
All clear things
are intelligible <--------
Logic is clear.
Logic is intelligible.
2. All these tables must be beautiful, because
they are green.
3. Because no drug addict is trustworthy, no
abnormal persons are trustworthy.
4. All things that are clear are appealing and
logic is appealing.
5. No trees are birds and trees are green.
6. No drug addict is trustworthy since no
abnormal persons are trustworthy.
7. Propaganda is not truthful because it is
8. All alcoholics are short-lived; therefore
Jim won’t live long.
9. Smith is a novelist. He must be a writer.
10. Since logic puzzles me, it is not
11. War is evil since it is inhuman.
After you complete the syllogisms, proceed to
determine whether they are valid or invalid.
The Golden Sheets, Set 2
GOLDEN SHEETS, SET 2
RULES OF THE FIGURES
(Based on the book An
Introduction to Logic, by H. W. B. Joseph, Oxford University
Rule 1: The minor premise must be
If it were negative then the conclusion would
be negative, and then the major term would be distributed. If so,
then the major premise would be negative, but from 2 negative
premises no conclusion follows. Therefore, the minor premise
cannot be negative.
Rule 2: The major premise must be
Since the middle term cannot be distributed
in the minor premise it must be distributed in the major premise,
according to the 1st rule of validity of the syllogism.
Rule 1: The conclusion must be negative.
In order to distribute the middle term,
according to the 1st rule of validity, one of the
premises must be negative, which implies that the conclusion must
Rule 2: The major premise must be
If the conclusion is negative, then the major
term is distributed, and according to the 2nd rule of
validity, it must be distributed in the major premise which
implies that this premise must be universal.
Rule 1: The minor premise must be
See explanation for Rule 1 in the First Figure.
Rule 2: The conclusion must be particular.
If the minor premise is affirmative, then the
minor term is not distributed which implies that it cannot be
distributed in the conclusion: this means that the conclusion must
Fourth Figure. Try to figure out the
explanations of these rules.
Rule 1: If either premise is negative, the
major premise must be universal.
Rule 2: If the major premise is
affirmative, the minor premise must be universal.
Rule 3: If the minor premise is
affirmative, the conclusion must be particular.
Golden Sheets, Set 2, Page 2
You should apply these rules to the 11 valid
moods in general. Doing so will produce the 19 essential moods
contained in the following poem:
Barbara Celarent Darii Ferio, first
Cesare Camestres Festino Baroco, second
Thrird, Darapti Disamis Datisi Felapton Bocardo Ferison
Fourth, Bramantip Camenes Dimaris Fesapo Fresison.
A few valid moods have not been included here
because their conclusions can be derived from the conclusions of
Golden Sheets, Set 2, Page 3,
EXERCISES p. 1
Rewrite the following syllogistic enthymemes
in standard form (putting the major premise first, the minor
premise second and the conclusion last). The missing component
must be indicated. After the arguments are completed, then
decide if they are valid or invalid.
1. All criminals are abnormal, thus no
trustworthy person is a criminal.
2. All apples are fruits and all fruits have
3. On the ground that pleasure is good, it is
sought by all human beings.
4. Roberto is a good father because good
husbands are good fathers.
5. Since no honest moralists are good
citizens, no good citizens practice what they preach.
6. Logic is unintelligible and no
unintelligible things are clear.
7. Smita is not a painter, because she does
not imitate nature.
8. All poets are artists; therefore Mr. Chang
is not an artist.
9. Since no tool is a cup, cups are not
10. All Socialists believe in sharing wealth;
thus all Communists are Socialists.
11. She can’t have a phone, since she is not
listed in the directory.
12. Since war is an evil, it should be
13. Normal people are trustworthy and no drug
addict is trustworthy.
14. Given that dolphins are mammals, no
dolphins are fish.
15. Pleasure is good since it is sought by all
16. Smita is a member of the union. She must
be a communist.
17. All good citizens practice what they
preach and good citizens are honest moralists.
18. Logic is not unintelligible on the ground
that it is clear.
Golden Sheets, Set 2, Page 4,
EXERCISES p. 2
Decide whether the following syllogistic
arguments are valid or invalid.
1. All mushrooms are fungi, and all mildews
are fungi; therefore all mushrooms are mildews.
2. Some teachers are married men; for some
teachers are bachelors, and no bachelors are married men.
3. All nudists are suntanned people; so all
nudists are nature lovers, for all suntanned people are nature
4. All red bugs are chiggers, and no fleas
are red bugs; hence no chiggers are fleas.
5. Some lawyers are not dishonest people, but
all judges are lawyers; so some judges are not dishonest people.
6. Some crimes of passion are not voluntary
actions; therefore no moral acts are crimes of passion, since all
moral acts are voluntary actions.
7. Some cenobites are not gold prospectors,
for no gold prospectors are gregarious, and some cenobites are not
8. Since no military generals are circus
midgets, and all circus midgets are undersized people, no
undersized people are military generals.
9. No theological hierarchies are
democracies; so, since all hierarchies are theological
hierarchies, all hierarchies are democracies.
10. Some atheists are obscure people, and no
obscure people are famous people; consequently some atheists are
11. All gin drinks are alcoholic beverages;
for all alcoholic beverages are intoxicating beverages, and all
gin drinks are intoxicating beverages.
12. No members of the Diogenes Club are
members of the Aristotelian Society; so no people interested in
philosophy arc members of the Diogenes Club, since all members of
the Aristotelian Society are people interested in philosophy.
13. No retired gamblers are good insurance
risks; so. since some jet pilots are not good insurance risks,
some jet pilots are not retired gamblers.
14. No band conductors are ballet dancers,
because some ballet dancers are not musically inclined people and
all band conductors are musically inclined people.